Eta invariant on articulated manifolds
Spectral Invariants on Non-compact and Singular Spaces
CRM Montreal
Richard Melrose
Department of Mathematics
Massachusetts Institute of Technology
24 July, 2012
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
1 / 28
Outline
1
Conjectures
2
Basics
3
History
4
General case
5
Distributions on the collective boundary
6
Boundary map
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
2 / 28
Introduction
I want to talk today about manifolds with corners. This may come
as no great surprise to many of you, but I suspect that I have not
talked enough about their basic geometry and analysis.
In this talk I will concentrate on incomplete metrics and the
corresponding Dirac operators.
In fact I will start by (roughly) stating two related conjectures.
That there should be such conjectures is well-known but perhaps
they have not often been stated precisely (and maybe for good
reason . . . ).
I want to at least show you that the tools now exist to check
whether these are true or not.
Maybe someone here would like to take up the challenge.
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
3 / 28
Conjectures
Conjecture (Eta invariant)
Let Y be an odd-dimensional articulated manifold without boundary
and suppose ð0 is an articulated Dirac operator on a unitary Clifford
module, V0 , with respect to a smooth incomplete metric then
ð0 : H 1 (Y ; V0 ) −→ L2 (Y ; V0 ) is self-adjoint with discrete spectrum and
the associated eta function and eta invariant are well-defined.
For this to make any sense I need to describe what
An articulated manifold Y is
An articulated Dirac operator on it is
Why it might be true.
The case that I do assert that this is true is when Y has articulation of
codimension one.
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
4 / 28
Conjectures
APS package
Conjecture (APS boundary condition)
Let X be an even-dimensional manifold (with corners) and suppose ð
is a Dirac operator on a unitary (Z2 -graded) Clifford module, V , with
respect to a smooth incomplete
metric then ð+ induces an articulated
Dirac operator ð0 on V0 = V ∂X and
n
o
3
1
ð+ : u ∈ H 2 (X ; V+ ); Π+ (ð0 )(u ∂X ) = 0 −→ H 2 (X ; V− )
is Fredholm with index given by
Z
b Ch0 (V ) + R − η(ð0 ).
A
ind(ð+ ) =
X
Here R is supposed to be the sum of integrals of a local differential
expressions on the boundary faces. I believe this to be true in
codimension two as I will explain below.
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
5 / 28
Basics
Manifolds (with corners)
Here is an extrinsic definition, correct but bad. Of course this is really a
theorem, a properly defined manifold (with corners) can always be
embedded in this sense.
Definition
An embedded compact manifold (with corners) X is a closed subset of
a compact manifold without boundary M of the form
X = {p ∈ M; ρi (p) ≥ 0 ∀ i ∈ {1, . . . , N}}
where ρi ∈ C ∞ (M) are real-valued functions such that for any
I ⊂ {1, . . . , N} and any p ∈ M
ρi (p) = 0 ∀ i ∈ I =⇒ ρi (p) are independent inTp∗ M, i ∈ I.
An (incomplete) metric on X is then by definition the restriction to X of
a metric on M. The same is true for bundles, differential operators etc.
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
6 / 28
Basics
Articulated manifolds
Here is a similar, perhaps even worse definition.
Definition
A compact articulated manifold without boundary is a (finite union of)
component(s) of the boundary of a compact manifold.
Again this is really a theorem, that an intrinically defined
articulated manifold can be embedded in this way.
So an articulated manifold is really a finite collection of compact
manifolds (with corners of course) with their boundary
hypersurfaces identified and consistently in higher codimension.
The absence of boundary is a completeness condition – there are
no unmatched hypersurfaces.
The important point is that an articulated manifold is a wobbly
thing – there are no angles between boundary hypersurfaces or
anything like that.
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
7 / 28
History
50 years ago – Atiyah and Singer
For an even-dimensional compact manifold without boundary, a
Dirac operator ð+ : C ∞ (X ; V+ ) −→ C ∞ (X ; V− ) is an elliptic
differential operator of first order, so Fredholm:
Nul(ð+ ) ⊂ C ∞ (X ; V+ ), Nul(ð− ) = (Ran(ð+ ))⊥
are finite-dimensional.
The index is computable:Z
ind(ð+ ) = dim Nul(ð+ ) − dim Nul(ð− ) =
b Ch0 .
A
X
In fact in this form, with the twisting Chern character of the Clifford
module, the index theorem is due to Berline, Getzler and
Vergne[4].
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
8 / 28
History
35 years ago – Atiyah, Patodi and Singer
For a Dirac operator on an odd-dimensional compact manifold, the
eta invariant, is well-defined in terms of the heat kernel by
Z ∞ 1
1
η(ð0 ) = √
Tr t − 2 ð0 ext(−itð20 ) dt.
π 0
A Dirac operator on a compact even-dimensional manifold with
boundary induces a self-adjoint Dirac operator on the boundary;
let Π+ (ð0 ) be the projection onto its positive part.
The operator with APS boundary condition
ð+ : u ∈ C ∞ (X ; V+ ); Π+ (ð+ )(u ∂X ) = 0 −→ C ∞ (X ; V− )
is Fredholm with index
Z
indAPS (ð+ ) =
b Ch0 +R − η(ð0 ).
A
If the operator is a product to first order at the boundary, R = 0.
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
9 / 28
History
Calderón’s sequence
The work of Calderón on boundary problems gives a very clean
approach to understanding the APS theorem.
Suppose given a linear, elliptic differential operator with smooth
coefficients on a compact manifold with boundary
D : C ∞ (X ; V+ ) −→ C ∞ (X ; V− ).
I will assume that all bundles carry inner products and that a
metric has been chosen
In particular D has a formal adjoint
D ∗ : C ∞ (X ; V− ) −→ C ∞ (X : V+ ).
Let C˙∞ (X ; V ) ⊂ C ∞ (X ; V ) be the closed subspace of elements
which vanish in Taylor series at the boundary then
Nul(D; C ∞ )
/ C ∞ (X ; V+ )
D
/ C ∞ (X ; V− )
/ Nul(D ∗ ; C˙∞ )
is exact with Nul(D ∗ ; C˙∞ ) = Nul D ∗ : C˙∞ (X ; V− ) −→ C˙∞ (X ; V+ ))
.
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
10 / 28
History
Calderón projector
The null space of the restriction to the boundary of smooth
solutions in the interior is finite dimensional
Nul(D; C˙∞ )
/ Nul(D; C ∞ )
|∂X
/ C ∞ (∂X ; V+ ).
Calderón showed that there is a projection precisely onto the
range of this restriction which is a pseudodifferential operator
ΠC ∈ Ψ0 (∂X ; V+ ), ΠC : C ∞ (∂X ; V+ ) −→ Nul(D; C ∞ ) .
∂X
For instance this is the case for the self-adjoint projection with
respect to a choice of metrics and inner products.
For any choice,
Ran(σ0 (ΠC )) = Ran+ (σ1 (D0 )),
the range of the symbol is always the span of the generalized
eigenvectors of the symbol of D0 in the right half plane where
D = N(∂x − iD0 ) at ∂X ; D0 ∈ Diff1 (∂X ; V ), x = 0 at ∂X .
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
11 / 28
History
Jumps formula – boundary case
Consider the null space on extendible distributions on M
Nul(D; C −∞ ) = {u ∈ C −∞ (X ; V+ ); Du = 0},
C −∞ (X ; V+ ) = C˙∞ (X ; V+ )0 .
Partial hypoellipticity up to the boundary implies that the restriction
to the boundary is well-defined (as are higher normal derivatives),
Nul(D; C −∞ ) 3 u 7−→ Bu = u ∂X ∈ C −∞ (∂X ; V+ ).
The ‘jumps formula’ is also a consequence of this:- There is a
unique v ∈ C −∞ (X ; V+ ) such that
v = 0 in M \ x, v = u on X \ ∂X
Pv = wδ(ρ) and w = −iσ(D)(dρ)(Bu).
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
12 / 28
History
Jumps and projector – boundary case
Now assume (for simplicity) that D = ð+ , is the restriction of a
Dirac operator on the whole of M ⊃ X and that
ð : C ∞ (M; V+ ) −→ C ∞ (M; V− ) is an isomorphism.
Then we get an explicit Calderón projector as
ΠC
C ∞ (∂X ; V+ ) 3 v
−iσ(D)(dρ)v ⊗ δ(ρ)
/ ΠC v ∈ C −∞ (∂X ; V+ )
O
B
ð−1
/ C −∞ (M; V+ )
|X \∂X
/ NulX (ð+ ; C −∞ )
In the general case one needs only do a little more work.
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
13 / 28
General case
I want to try to convince you of the existence of such a picture in
the general case of a compact manifold with (non-trivial) corners.
The spaces C˙∞ (X ; V ) with dual C −∞ (X ; V ) and C ∞ (X ; V ) with
dual C˙−∞ (X ; V ) are well-defined (metrics everywhere) and in
terms of an extension X ⊂ M
C˙∞ [resp C˙−∞ ](X ; V ) = {u ∈ C ∞ [resp C −∞ ](X ; V ); supp(u) ⊂ X )
C ∞ [resp C −∞ ](X ; V ) = C ∞ [resp C −∞ ](M; V )
.
X \∂X
So let ð+ : C ∞ (X ; V+ ) −→ C ∞ (X ; V− ) be a Dirac operator, this
makes all the pesky finite-dimensional Nul(ð± ; C˙∞ ) trivial.
In particular surjectivity holds
Nul(ð+; C −∞ )
/ C −∞ (X ; V+ )
D
/ / C −∞ (X ; V− )
So the whole issue is to define B and ΠC .
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
14 / 28
General case
Although partial hypoellipticity fails we can still use a variant of the
jumps formula to define B.
There is a surjective restriction map
C˙−∞ (M; V ) −→ C −∞ (M; V )
with null space the distributions supported by the boundary;
u ∈ Nul(ð+ ; C −∞ ) can be extended to M to vanish outside X .
In fact there is always such a ‘zero extension’ v ∈ C˙−∞ (X ; V+ ) with
X
ð+ (v ) =
vH ⊗ δ(ρH ), vH ∈ C˙−∞ (H; V− )
(1)
H
Here, each boundary hypersurface H has a defining function ρH
and the space on the right is a well-defined in C˙−∞ (X ; V− ).
However, there are two problems, the zero extension – even with
this property – is not unique and nor are the ‘boundary values’ vH
(even fixing the ρH which we can. So the presentation (1) is also
not unique; the crucial question is just how non-unique.
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
15 / 28
General case
Formal boundary data
To answer this we now switch to the ‘formal smooth theory’.
Think of ∂X as an articulated manifold – the union of the boundary
hypersurfaces with only their boundaries identified in the obvious
way. Then the ‘smooth’ sections of a bundle over ∂X are
n
o
C ∞ (∂X ; V ) = ui ∈ C ∞ (Hi ; V ); ui H ∩H = uj H ∩H = C ∞ (M; V )∂X .
i
j
i
j
As remarked above, this space is ‘too big’ in the sense that there
are no compatibility conditions for the normal derivatives at
intersections of boundary faces.
However, a first order elliptic differential operator, gives rise to
much smaller subspace of ‘compatible’ sections
C ∞ (∂X ; V+ ) = {u ∈ C ∞ (X ; V+ ); Du ∈ C˙∞ (X ; V+ )}
D
∂X
⊂ C ∞ (Y ; V+ ).
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
16 / 28
General case
Properties of CD∞ .
Lemma
For an elliptic differential operator on a compact manifold (with
corners) D ∈ Diff1 (X ; V+ , V− ) restriction to any one of the of the
boundary hypersurfaces defines a surjective map
H
CD∞ (∂X ; V+ )
/ / C ∞ (H; V+ ), H ∈ M1 (M),
and there is a natural extension giving an injective map
L
˙∞ (H; V+ ) 
H∈M1 (M) C
/ C ∞ (∂X ; V+ ).
D
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
(2)
24 July, 2012
17 / 28
Distributions on the collective boundary
Note that ∂X can be ‘smoothed’ (more like annealed!) to a
compact manifold without boundary
Y
ρH = }, > 0 small.
H̃ = {p ∈ X ;
H
Then CD∞ (∂X ; V+ ) ‘looks’ like C ∞ (H̃; V ) in the sense that the
Taylor series at any boundary point coming from one boundary
hypersurface determines the Taylor series at any others.
This new space is not a module of C ∞ (∂X ).
On the other hand, it does have a topology very similar to that of
C ∞ (H̃; V ) such that the maps in (2) are continuous.
The dual space C −∞ (∂X ; V+ ) is similar to C −∞ (H̃; V+ ).
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
18 / 28
Distributions on the collective boundary
Properties of CD−∞ .
Lemma
The topological dual CD−∞ (∂X ; V+ ) comes equipped with a natural
surjection to extendible distributions on the boundary hypersurfaces
CD−∞ (∂X ; V+ )
/
L
H∈M1 (X ) C
−∞ (H; V
+)
and injections on supported distributions for each H ∈ M1 (M)

C˙−∞ (H; V+ ) / C −∞ (∂X ; V+ )
D
such that the collective map is surjective
[·] :
L
˙−∞ (H; V+ )
H∈M1 (X ) C
/ / C −∞ (∂X ; V+ ).
D
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
19 / 28
Boundary map
This space answers the question of just how well-defined the boundary
data for the null space of an elliptic operator on a compact manifold
with corners is where now we have a boundary pairing which gives
CD−∞ (∂X ; V+ ) = (CD∞∗ (∂X ; V− ))0 .
Theorem
With the global hypotheses above on the first order elliptic differential
operator D, there is a well-defined injective boundary map B giving a
commutative diagram
Nul(D; C −∞ )
B
/ C −∞ (∂X ; V+ )
D
(
)
˙−∞
v ∈C
(X ; V+ ), ð+ v =
X
−iσ(D)(dρJ )wH ⊗ δ(ρH )
7−→ [wH ]
H
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
20 / 28
Boundary map
Calderón projector, corners case
This in turn allows us to define the Calderón projector as in the case of
a manifold with boundary except for the extra algebraic overhead
ΠC : CD−∞ (∂X ; V+ ) −→ CD−∞ (∂X ; V+ ) by
!
ΠC ([wH ]) = B
D
−1
(
X
−iσ(dρH )wH δ(ρH ))X
.
H
Theorem
The Calderón projector is a continuous projection on CD−∞ (∂X ; V+ ) and
has range precisely equal to the range of B which maps Nul(D; C −∞ )
injectively into CD−∞ (∂X ; V+ ).
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
21 / 28
Boundary map
This Calderón projector is as close to being a pseudodifferential
operator as one could expect on an articulated manifold. Namely,
it consists of pseudodifferential operators on each of the
hypersurfaces plus ‘Poisson’ type operators between them.
In particular, it preserves CD∞ (∂X ; V+ ), even though the
pseudodifferential pieces do not satisfy the transmission condition.
The singularities are cancelled by the Poisson pieces.
These results should extend to the general case where D is not
assumed to either have the extension property or the unique
continuation property.
The extension to higher order systems would be a more serious
pain!
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
22 / 28
Boundary map
Continuing under the global assumptions, observe that for t ∈ R,
|t| < 12 , and on any compact manifold with corners, the extendible
and supported Sobolev spaces are identified
Ḣ t (H; V ) = (H −t (H; V ))0 ≡ H t (H; V ), −
1
1
<t < .
2
2
That is, each element of these Sobolev spaces has a unique zero
extension with the same regularity (with which it can therefore be
identified).
In view of the properties of the spaces discussed above it follows
that
M
1
1
H t (H; V− ) ⊂ CD−∞ (∂X ; V+ ), − < t <
2
2
H∈M1 (X )
are well-defined subspaces for any elliptic first-order D.
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
23 / 28
Boundary map
The regularity properties of D −1 show that that
M
ΠC acts on
H t (H; V+ ), −
H∈M1 (M)
1
1
<t < ,
2
2
with range precisely the boundary restrictions of
1
Nuls (D) = {u ∈ H s (X ; V+ ); Du = 0}, s = t + .
2
Thus, for instance, for
1
1
2
< s < 1 there is a short exact sequence
{U ∈ H s− 2 (∂X ; V+ ); ΠC U = U}
/ H s (X ; V+ )
D
/ H s−1 (H; V− ).
where the first map is a Poisson operator.
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
24 / 28
Boundary map
For Dirac operators ‘restriction’ to a boundary hypersurface is
functorial - giving a Dirac operator ðH on each H ∈ M1 (X ).
This involves the product decomposition near a hypersurface in
terms of the distance, in which the metric decomposes as
g = dx 2 + x 2 h(x), h(x) a family of metric on H.
There is no (simple) analogue of this in codimension two.
Nevertheless the double restriction, from ð+ on X to a boundary
face of codimension two is consistent (with change of orientation)
(ðH )H∩G + (ðG )H∩G = 0.
(1)
This is what is meant above by a Dirac operator on an articulated
manifold – on each boundary hypersurface there is a Dirac
operator ðH associated to a metric and a Clifford module (and
unitary Clifford connection). The bundles and metrics must be
consistent on the intersection faces of codimension two – from
either side one gets the same restriction – and the Clifford
modules are consistent in the sense of (1).
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
25 / 28
Boundary map
This is enough to give sense to the ‘Eta invariant’ conjecture.
Conjecture (Eta invariant)
Let Y be an odd-dimensional articulated manifold without boundary
and suppose ð0 is an articulated Dirac operator on a unitary Clifford
module, V0 , with respect to a smooth incomplete metric then
ð0 : H 1 (Y ; V0 ) −→ L2 (Y ; V0 ) is self-adjoint with discrete spectrum and
the associated eta function and eta invariant are well-defined.
I claim this is true for an articulated manifold with intersection
faces only of codimension one – this is close to the boundary
case.
One can get a parametrix, in the sense of an inverse modulo
compact errors by summing the generalized inverse of the APS
problem on each boundary hypersurface (there is an odd/even
switch here).
In particular the projection onto the positive part makes sense.
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
26 / 28
Boundary map
APS package
Conjecture (APS boundary condition)
Let X be an even-dimensional manifold (with corners) and suppose ð
is a Dirac operator on a unitary (Z2 -graded) Clifford module, V , with
respect to a smooth incomplete
metric then ð+ induces an articulated
Dirac operator ð0 on V0 = V ∂X and
n
o
3
1
ð+ : u ∈ H 2 (X ; V+ ); Π+ (ð0 )(u ∂X ) = 0 −→ H 2 (X ; V− )
is Fredholm with index given by
Z
b Ch0 (V ) + R − η(ð0 ).
A
ind(ð+ ) =
X
The existence of Π+ follows from the discussion above in case X has
boundary of codimension two. The Fredholm property should follow
from a symbolic analysis of the two projections, Calderón and APS.
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
27 / 28
Boundary map
Final remarks
A lot of this is conjectural, but the case of X of codimension two is
surely within reach.
There is the possibility of induction over boundary codimension.
If this is all too easy for you, try the ‘annealing limit’ as ↓ 0,
passing from a manifold with boundary to the general case.
I have not given references but there is a large literature related to
this subject – but not the Calderón projector as far as I know.
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
28 / 28
Boundary map
M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry
and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc.
77 (1975), 43–69. MR 53 #1655a
, Spectral asymmetry and Riemannian geometry. II, Math.
Proc. Cambridge Philos. Soc. 78 (1975), no. 3, 405–432. MR 53
#1655b
, Spectral asymmetry and Riemannian geometry. III, Math.
Proc. Cambridge Philos. Soc. 79 (1976), no. 1, 71–99. MR 53
#1655c
Nicole Berline, Ezra Getzler, and Michèle Vergne, Heat kernels
and Dirac operators, Springer-Verlag, Berlin, 1992.
Richard Melrose ( Department of Mathematics Massachusetts
Eta invariant Institute
on articulated
of Technology
manifolds)
24 July, 2012
28 / 28